Problem: Determine how many solutions exist for the system of equations. ${-5x-y = 6}$ ${5x+y = -6}$
Answer: Convert both equations to slope-intercept form: ${-5x-y = 6}$ $-5x{+5x} - y = 6{+5x}$ $-y = 6+5x$ $y = -6-5x$ ${y = -5x-6}$ ${5x+y = -6}$ $5x{-5x} + y = -6{-5x}$ $y = -6-5x$ ${y = -5x-6}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -5x-6}$ ${y = -5x-6}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${-5x-y = 6}$ is also a solution of ${5x+y = -6}$, there are infinitely many solutions.